2.8 Appendix
3.1.4 Notation : semi-classical operators and geometrical setting
3.1.4.1 Semi-classical operators on Rd
We shall use of the notationhηi := (1 +|η|2)12. For a parameter h∈ (0, h0] for some h0 >0, we denote bySm(Rd×Rd),Sm for short, the space of smooth functionsa(z, ζ, h) that satisfy the following property : for allα,β multi-indices, there existsCα,β≥0, such that
∂αz∂ζβa(z, ζ, h)≤Cα,βhζim−|β|, z∈Rd, ζ∈Rd, h∈(0, h0].
Then, for all sequencesam−j∈Sm−j,j∈N, there exists a symbola∈Smsuch thata∼P
jhjam−j, in the sense that
a− P
j<N
hjam−j∈hNSm−N (3.17)
(see for instance [Mar02, Proposition 2.3.2] or [H¨or85, Proposition 18.1.3]), with am as principal symbol. We define Ψm as the space of semi-classical operators A = Op(a), for a ∈ Sm, formally defined by
A u(z) = (2πh)−d ZZ
eihz−t,ζi/ha(z, ζ, h)u(t)dt dζ, u∈S′(Rd).
We shall denote the principal symbol am byσ(A). We shall use techniques of pseudo-differential calculus in this article, such as construction of parametrices, composition formula, formula for the symbol of the adjoint operator, etc. We refer the reader to [Tay81, H¨or85, Mar02]. We provide composition and change of variables formula.e. in the case of tangential operators in Appendix 3.7.
Those formula.e. can be adapted to the case of operators acting in the whole spaceRd. In the main text the variablez will be (x0, x)∈Rn+1 andζ= (ξ0, ξ)∈Rn+1.
We set
S−∞= T
m>0
S−m, h∞S−∞= T
m>0
hmS−m, Ψ−∞= T
m>0
Ψ−m, h∞Ψ−∞= T
m>0
hmΨ−m.
Note that if there exists a closed set F such that in the asymptotic expansion (3.17) we have supp(am−j)⊂F, j∈N, then a representative ofamoduloh∞S−∞ can be chosen supported inF. We shall also denote byDmthe space of semi-classicaldifferentialoperators, i.e., the case where a(z, ζ, h) is a polynomial function of ordermin ζ. In particular we set
D=h
i∂, and we have σ(D) =ξ.
We now introduce Sobolev spaces onRd and Sobolev norms which are adapted to the scaling parameter h. The natural norm onL2(Rd) is written as kukL2(Rd)=kuk0 := (R
|u(x)|2 dx)12. Let r∈R; we then set
kukr=kukHr(Rd)=kΛruk0, with Λr:= Op(hξir) and Hr(Rd) :={u∈S′(Rd); kukr<∞}. The spaceHr(Rd) is algebraically equal to the classical Sobolev spaceHr(Rd). For a fixed value of h, the normk.kr is equivalent to the classical Sobolev norm that we writek.kHr(Rd). However, these norms are not uniformly equivalent ashgoes to 0.
3.1.4.2 Tangential semi-classical operators on Rd,d≥2
We set z = (z′, zd), z′ = (z1, . . . , zd−1) and ζ′ = (ζ1, . . . , ζd−1) accordingly. We denote by STm(Rd×Rd−1), STm for short, the space of smooth functions b(z, ζ′, h), defined for h∈ (0, h0] for
some h0>0, that satisfy the following property : for all α, β multi-indices, there exists Cα,β ≥0, such that
∂zα∂βζ′b(z, ζ′, h)≤Cα,βhζ′im−|β|, z∈Rd, ζ′∈Rd−1, h∈(0, h0].
As above, for any sequence bm−j ∈ STm−j, j ∈ N, there exists a symbol b ∈ STm such that b ∼ P
jhjbm−j, in the sense thatb−P
j<Nhjbm−j∈hNSmT−N, withbmas principal symbol. We define ΨmT as the space of tangential semi-classical operatorsB = OpT(b) (observe the notation we adopt is different from above to avoid confusion), forb∈STm, formally defined by
B u(z) = (2πh)−(d−1) ZZ
eihz′−t′,ζ′i/hb(z, ζ′, h)u(t′, zd)dt′ dζ′, u∈S′(Rd).
In the main text the variablezwill be (x0, x′, xn)∈Rn+1 andζ′ = (ξ0, ξ′)∈Rn.
We shall also denote the principal symbolbmbyσ(B). In the case where the symbol is polynomial inζ′andh, we shall denote the space of associated tangentialdifferentialoperators byDm
T . We shall denote by ΛsT the tangential pseudo-differential operator whose symbol ishζ′is. We set
ST−∞= T
m>0
ST−m, h∞ST−∞= T
m>0
hmS−Tm, Ψ−∞T = T
m>0
Ψ−Tm, h∞Ψ−∞T = T
m>0
hmΨ−Tm.
For function defined onzd = 0 or restricted to zd = 0, following [LR95, LR97], we shall denote by (., .)0 the inner product, i.e., (f, g)0 :=RR
f(z′)g(z′)dz′. The induced norm is denoted by |.|0, i.e.,|f|20= (f, f)0. Forr∈Rwe introduce
|f|Hr(Rd−1)=|f|r:=|ΛrTf|0. (3.18) The composition Formula and the action of change of variables are given in Appendix 3.7.1.
Note that we shall keep the notation ΨmT for operators with symbols independent ofzd, acting on{zd = 0}. These operators are in fact in Ψm(Rd−1). A similar notation will be used in the case of operators on a manifold.
3.1.4.3 Local charts, pullbacks, and Sobolev norms
The submanifoldS is of dimensionn−1 and is furnished with afiniteatlas (Uj, φj),j∈J. The mapsφj :Uj →U˜j⊂Rn−1is a smooth diffeomorphism. If Uj∩Uk 6=∅ we also set
φjk: φj(Uj∩Uk)⊂U˜j→φk(Uj∩Uk)⊂U˜k, y7→φk◦φ−j1(y).
The local charts and the diffeomorphisms we introduce are illustrated in Figure 3.2.
For a diffeomorphismφbetween two open sets,φ:U1→U2, the associated pullback (here stated for continuous functions) is
φ∗:C(U2)→C(U1), u7→u◦φ.
For a function defined on phase-space, e.g. a symbol, the pullback is given by φ∗u(y, η) =u(φ(y),t φ′(y)−1
η), y∈U1, η∈Ty∗(U1), u∈C(T∗U2). (3.19) We shall use semi-classical Sobolev norms over the manifold S together with a finite atlas (Uj, φj)j,φj:Uj→Rn−1, and a partition of unity (ψj)j subordinated to this covering ofS :
ψj ∈C∞(S), supp(ψj)⊂Uj, 0≤ψj≤1, P
j
ψj = 1.
3.1. Introduction 105
φ
jkφ
kS
φ
jU ˜
k,jU ˜
kU
jU
kU ˜
jRn−1 Rn−1
U ˜
j,kFigure3.2 – Local charts and diffeomorphisms for the submanifoldS.
We then set :
|u|Hr(S)=P
j | φ−j1∗
ψju|Hr(Rn−1). (3.20)
Note that the l.h.s. denotes a norm on the manifold and the r.h.s. is defined in (3.18). We shall need the following elementary result.
Lemma 3.8. Let (fj)j be a family of smooth functions on S with supp(fj) ⊂ Uj and P
jfj = f ≥ C > 0 in S. We set Nr(u) = P
j| φ−j1∗
fju|Hr(Rn−1). Then Nr is an equivalent norm to
|.|Hr(Rn−1), uniformly inh.
For a proof see Appendix 3.8.1. Note that theL2-norm (r= 0) defined in (3.20) is equivalent to the naturalL2-norm on the Riemannian manifold S given through the inner product in (3.5).
Norms in codimension 1. For a functionudefined on (0, X0) ×Rn−1we set
|u|0=|u|L2((0,X0)×Rn−1), |u|21=|Dx0u|20+ Z X0
0 |u|2H1(Rn−1)dx0.
Note that the latter norm is equivalent to |u|H1(R×Rn−1) if moreover the function uis compactly supported in the x0 variable. For a functionudefined on (0, X0) ×S, we set
|u|ℓ=P
j | φ−j1∗
ψju|ℓ, ℓ= 0,1, (3.21)
whereφj stands for Id⊗φj.
Norms in all dimensions. For a functionudefined on (0, X0) ×Rn−1×Rwe set kuk0=kukL2((0,X0)×Rn−1×R), kuk21=kDx0uk20+
Z X0
0
Z
Rkuk2H1(Rn−1)dx0dxn+kDxnuk20. Note that the latter norm is equivalent tokukH1(R×Rn−1×R)if moreover the functionuis compactly support in the x0 variable. For a functionudefined on (0, X0) ×S×R, we set
kukℓ=P
j k φ−j1∗
ψjukℓ, ℓ= 0,1, (3.22)
whereφj stands for Id⊗φj⊗Id.
The following lemma is a counterpart of Lemma 3.8 when working on a local chart of (0, X0)×S or (0, X0)×S×R.
Lemma 3.9. Let u be such that supp(u) ⊂ K ⊂ (0, X0)×Uj (resp. (0, X0)×Uj×R) with K compact. Then for some constant CK we have
CK−1|u|ℓ≤ | φ−j1∗
u|ℓ≤CK|u|ℓ (resp.CK−1kukℓ≤ k φ−j1∗
ukℓ≤CKkukℓ), ℓ= 0,1.
Proof. We treat the case of a function defined in (0, X0)×Uj. Consider a partition of unity ofS, P
kψˆk= 1, ˆψk∈C∞
c ((0, X0)×Uk), such that 1⊗ψˆj= 1 in a neighborhood ofK. Then the induced norms are equivalent to that given above by Lemma 3.8 and for the particular functionuthey are equal to| φ−j1∗
u|ℓ,ℓ= 0,1.
Tangential semi-classical operators on a manifold. We can define tangential semi-classical operators on a manifold by means of local representations. This relies on the change of variables formula for semi-classical operators in Rd presented in Appendix 3.7.1. We provide details of this construction in Appendix 3.7.2. In particular we define the local symbol of the operator in each chart and its principal symbol on the manifold. We also provide composition and Sobolev regularity results for such operators. In Section 3.3.6 below we introduce a particular class of tangential operators that will be important in the proof of the Carleman estimate as they will allow us to separate the analysis into microlocal regions.
A trace formula. In the sections below, we shall also use of the following trace formula [LR97, page 486] connecting the tangential and volume norms introduced above :
|ψ|xn=0+|0≤Ch−12kψk1, (3.23) forψdefined on Rn+1, as well as forψdefined on (0, X0)×S×[0,2ε].